EXPONENTIAL FUNCTION BASICS COMMON CORE ALGEBRA II BASIC EXPONENTIAL FUNCTIONS

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1 Name: Date: EXPONENTIAL FUNCTION BASICS COMMON CORE ALGEBRA II You studied eponential functions etensivel in Common Core Algebra I. Toda's lesson will review man of the basic components of their graphs and behavior. Eponential functions, those whose eponents are variable, are etremel important in mathematics, science, and engineering. BASIC EXPONENTIAL FUNCTIONS where Eercise #1: Consider the function the graph on the grid provided. 2. Fill in the table below without using our calculator and then sketch Eercise #2: Now consider the function and sketch the graph on the aes provided Using our calculator to help ou, fill out the table below

2 Eercise #3: Based on the graphs and behavior ou saw in Eercises #1 and #2, state the domain and range for an eponential function of the form. b Domain (input set): Range (output set): Eercise #4: Are eponential functions one-to-one? How can ou tell? What does this tell ou about their inverses? Eercise #5: Now consider the function 73 (a) Determine the -intercept of this function algebraicall. Justif our answer.. (b) Does the eponential function increase or decrease? Eplain our choice. (c) Create a rough sketch of this function, labeling its - intercept Eercise #6: Consider the function (a) How does this function s graph compare to that of 1 3? What does adding 4 do to a function's graph? (b) Determine this graph s -intercept algebraicall. Justif our answer. (c) Create a rough sketch of this function, labeling its - intercept. 2

3 EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY COMMON CORE ALGEBRA II Eponential functions are ver important in modeling a variet of real world phenomena because certain things either increase or decrease b fied percentages over given units of time. You looked at this in Common Core Algebra I and in this lesson we will review much of what ou saw. Eercise #1: Suppose that ou deposit mone into a savings account that receives 5% interest per ear on the amount of mone that is in the account for that ear. Assume that ou deposit $400 into the account initiall. (a) How much will the savings account increase b over the course of the ear? (b) How much mone is in the account at the end of the ear? (c) B what single number could ou have multiplied the $400 b in order to calculate our answer in part (b)? (d) Using our answer from part (c), determine the amount of mone in the account after 2 and 10 ears. Round all answers to the nearest cent when needed. (e) Give an equation for the amount in the savings account as a function of the number of ears since the $400 was invested. (f) Using a table on our calculator determine, to the nearest ear, how long it will take for the initial investment of $400 to double. Provide evidence to support our answer. The thinking process from Eercise #1 can be generalized to an situation where a quantit is increased b a fied percentage over a fied interval of time. This pattern is summarized below: INCREASING EXPONENTIAL MODELS If quantit Q is known to increase b a fied percentage p, in decimal form, then Q can be modeled b where represents the amount of Q present at and t represents time. Eercise #2: Which of the following gives the savings S in an account if $250 was invested at an interest rate of 3% per ear? (1) S t t (3) S (2) S t (4) S t 3

4 Decreasing eponentials are developed in the same wa, but have the percent subtracted, rather than added, to the base of 100%. Just remember, ou are ultimatel multipling b the percent of the original that ou will have after the time period elapses. Eercise #3: State the multiplier (base) ou would need to multipl b in order to decrease a quantit b the given percent listed. (a) 10% (b) 2% (c) 25% (d) 0.5% DECREASING EXPONENTIAL MODELS If quantit Q is known to decrease b a fied percentage p, in decimal form, then Q can be modeled b where represents the amount of Q present at and t represents time. Eercise #4: If the population of a town is decreasing b 4% per ear and started with 12,500 residents, which of the following is its projected population in 10 ears? Show the eponential model ou use to solve this problem. (1) 9,230 (3) 18,503 (2) 76 (4) 8,310 Eercise #5: The stock price of WindpowerInc is increasing at a rate of 4% per week. Its initial value was $20 per share. On the other hand, the stock price in GerbilEnerg is crashing (losing value) at a rate of 11% per week. If its price was $120 per share when Windpower was at $20, after how man weeks will the stock prices be the same? Model both stock prices using eponential functions. Then, find when the stock prices will be equal graphicall. Draw a well labeled graph to justif our solution. 4

5 COMPOUND INTEREST COMMON CORE ALGEBRA II In the worlds of investment and debt, interest is added onto a principal in what is known as compound interest. The percent rate is tpicall given on a earl basis, but could be applied more than once a ear. This is known as the compounding frequenc. Let's take a look at a tpical problem to understand how the compounding frequenc changes how interest is applied. Eercise #1: A person invests $500 in an account that earns a nominal earl interest rate of 4%. (a) How much would this investment be worth in 10 ears if the compounding frequenc was once per ear? Show the calculation ou use. (b) If, on the other hand, the interest was applied four times per ear (known as quarterl compounding), wh would it not make sense to multipl b 1.04 each quarter? (c) If ou were told that an investment earned 4% per ear, how much would ou assume was earned per quarter? Wh? (d) Using our answer from part (c), calculate how much the investment would be worth after 10 ears of quarterl compounding? Show our calculation. So, the pattern is fairl straightforward. For a shorter compounding period, we get to appl the interest more often, but at a lower rate. Eercise #2: How much would $1000 invested at a nominal 2% earl rate, compounded monthl, be worth in 20 ears? Show the calculations that lead to our answer. (1) $ (3) $ (2) $ (4) $ This pattern is formalized in a classic formula from economics that we will look at in the net eercise. Eercise #3: For an investment with the following parameters, write a formula for the amount the investment is worth, A, after t-ears. P = amount initiall invested r = nominal earl rate n = number of compounds per ear 5

6 The rate in Eercise #1 was referred to as nominal (in name onl). It's known as this, because ou effectivel earn more than this rate if the compounding period is more than once per ear. Because of this, bankers refer to the effective rate, or the rate ou would receive if compounded just once per ear. Let's investigate this. Eercise #4: An investment with a nominal rate of 5% is compounded at different frequencies. Give the effective earl rate, accurate to two decimal places, for each of the following compounding frequencies. Show our calculation. (a) Quarterl (b) Monthl (c) Dail Practice: 1. You deposit $10,000 in an account that pas 6% interest. Find the balance after 10 ears if the interest is compounded a) quarterl b) Monthl 2. $2000 is deposited in an account that pas 8% annual interest, compounded monthl. What is the balance after 5 ears? 3. A parent, following the birth of a child, wants to make an initial investment that will grow to $10,000 b the child s 20th birthda. Interest is compounded continuousl at 8%. What should that initial investment be? 4. Complete the table: Invest $1 for 1 ear at 100% compound interest and compare the result. Annuall Bi- Annuall Quarterl Monthl Weekl Dail Hourl Ever Minute Ever Second 6

Good eamples of important numbers are 0, 1, i, and. In this lesson ou will be introduced to an important number given the letter e for its inventor Leonhard Euler (1707-1783).

7 Conclusion: Quick write: Write down an smbols that have defined values. Wh are these smbols used instead of numbers? THE NUMBER e AND THE NATURAL LOGARITHM COMMON CORE ALGEBRA II There are man numbers in mathematics that are more important than others because the find so man uses in either mathematics or science. Good eamples of important numbers are 0, 1, i, and. In this lesson ou will be introduced to an important number given the letter e for its inventor Leonhard Euler ( ). This number plas a crucial role in Calculus and more generall in modeling eponential phenomena. THE NUMBER e 1. Like, e is irrational. 2. e 3. Used in Eponential Modeling Eercise #1: Which of the graphs below shows e? Eplain our choice. Check on our calculator. (1) (2) (3) (4) Eplanation: Ver often e is involved in eponential modeling of both increasing and decreasing quantities. The creation of these models is beond the scope of this course, but we can still work with them.we could compound at smaller and smaller frequenc intervals, eventuall compounding all moments of time. In our formula from Eercise #3, we would be letting n approach infinit. Interestingl enough, this gives rise to continuous compounding and the use of the natural base e in the famous continuous compound interest formula. CONTINUOUS COMPOUND INTEREST 7 For an initial principal, P, compounded continuousl at a nominal earl rate of r, the investment would be worth an amount A given b:

8 Eercise #5: A person invests $350 in a bank account that promises a nominal rate of 2% continuousl compounded. (a) Write an equation for the amount this investment would be worth after t-ears. (b) How much would the investment be worth after 20 ears? (c) Algebraicall determine the time it will take for the investment to reach $400. Round to the nearest tenth of a ear. (d) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Practice: A student wants to save $8,000 for college in four ears. How much should be put into an account that earns 5.2% annual interest compounded continuousl? 5. How long would it take to double our principal at an annual interest rate of 8% compounded continuousl? 8

ACTIVITY: Comparing Types of Growth

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